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Wigner entropy conjecture and the interference formula in quantum phase space

Zacharie Van Herstraeten1,2 and Nicolas J. Cerf3,1

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Phys. Rev. A 112, 062207 – Published 3 December, 2025

DOI: https://doi.org/10.1103/1ftk-dm7l

Abstract

Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states—called Wigner entropy for brevity—emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the p-norms of cross Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry under convolution exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter α≥1/2.

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Phys. Rev. A 112, 062207– Published 3 December, 2025
  • Received 26 March 2025
  • Accepted 25 June 2025
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